“RECENT DEVELOPMENTS IN FIXED POINT ANDBANACH SPACE THEORY.”

WORKSHOP IN HONOUR OF BRAILEY SIMS,

UNIVERSITY OF NEWCASTLE

SAT 22/AUGUST/2015, 9:30 AM

CHRIS LENNARD, UNIVERSITY OF PITTSBURGH

Date: August 19, 2015.

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Abstract

I will discuss three different recent results in fixed point and Banach spacetheory. Aspects of each were strongly influenced by Brailey’s life and career asan extremely fine person, mathematician, teacher and mentor.

Parts (1), (2) and (3) were all partially inspired by Brailey’s wonderful seminarseries at Kent State University in 1986, entitled: “The Existence Question forFixed Points of Nonexpansive Maps.” As a new Ph.D. student at KSU, meet-ing and interacting with Brailey during my first year was one of the highlightsof my Ph.D. studies - and my life, both generally and mathematically.

Further, my interest in characterizing the fixed point property in c0, whichled in part to Item (3), was initiated by a paper of Enrique Llorens-Fuster andBrailey: “The fixed point property in c0”, Canadian Mathematical Bulletin41(4) (1998), 413-422.

1. Part (1): Introduction

In 1965, Browder [B1] proved: [♠] [For every closed,bounded, convex (non-empty) subset C of a Hilbertspace (X, ‖·‖), for all nonexpansive mappings T : C → C

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[i.e., ‖Tx−Ty‖ ≤ ‖x− y‖, for all x, y ∈ C], T has a fixedfixed point in C.] Soon after, also in 1965, Browder[B2] and Gohde [G] (independently) generalized the re-sult [♠] to uniformly convex Banach spaces (X, ‖ · ‖);e.g., X = Lp, 1 < p <∞, with its usual norm ‖ · ‖p.Later in 1965, Kirk [K] further generalized [♠] to

all reflexive Banach spaces X with normal structure:those spaces such that all non-trivial closed, bounded,convex sets C have a smaller radius than diameter.

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Spaces (X, ‖ · ‖) with the property of Browder [♠] be-came known as spaces with the “fixed point propertyfor nonexpansive mappings” (FPP (n.e.)).Concerning Kirk’s theorem, we may ask what fur-

ther generalizations are possible? After 50 years, itremains an open question as to whether or not ev-ery reflexive Banach space (X, ‖ ·‖) has the fixed pointproperty for nonexpansive maps.Recently (in 2009), Domınguez Benavides [5] proved

the following intriguing result: [Given a reflexive Ba-nach space (X, ‖ · ‖), there exists an equivalent norm‖ · ‖∼ on X such that (X, ‖ · ‖∼) has the fixed pointproperty for nonexpansive mappings]. This improvesa theorem of van Dulst [8] for separable reflexive Ba-nach spaces.

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In contrast to this result, the non-reflexive Banachspace (`1, ‖ · ‖1), the space of all absolutely summablesequences, with the absolute sum norm ‖ · ‖1, fails thefixed point property for nonexpansive mappings. E.g.,let C := {(tn)n∈N : each tn ≥ 0 and

∑∞n=1 tn = 1}. This

is a closed, bounded, convex subset of `1. LetT : C → C be the right shift map on C; i.e.,

T (t1, t2, t3, . . . ) := (0, t1, t2, t3, . . . ) .

T is clearly ‖·‖1-nonexpansive (being an isometry) andfixed point free on C.

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Recently (in 2008), in another significant develop-ment, Lin [13] provided the first example of a non-reflexive Banach space (X, ‖ · ‖) with the fixed pointproperty for nonexpansive mappings. Lin verified thisfact for (`1, ‖ · ‖1) with the equivalent norm ‖| · ‖| givenby

‖|x‖| = supk∈N

8k

1 + 8k

∞∑n=k

|xn|, for all x = (xn)n∈N ∈ `1 .

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What about (c0, ‖ · ‖∞), the Banach space of real-valued sequences that converge to zero, with the ab-solute supremum norm ‖ · ‖∞? This is another non-reflexive Banach space of great importance in Banachspace theory. It also fails the fixed point property fornonexpansive mappings. E.g., let

C := {(tn)n∈N : each tn ≥ 0, 1 = t1 ≥ t2 ≥ · · · ≥ tn ≥ tn+1

−→ 0, as n→∞} .

Let U : C → C be the natural right shift map.

U(t1, t2, t3, . . . ) := (1, t1, t2, t3, . . . ) .

Then U is a ‖ · ‖∞-nonexpansive (isometric, actually)map with no fixed points in the closed bounded convexset C.

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It is natural to ask whether there is a c0-analogue ofLin’s theorem about `1. It remains an open questionas to whether or not there exists an equivalent norm‖ · ‖∼ on (c0, ‖ · ‖∞) such that (c0, ‖ · ‖∼) has the fixedpoint property for nonexpansive mappings. However,if we weaken the nonexpansive condition to “asymp-totically nonexpansive”, then the answer is “no”. In2000, Dowling, Lennard and Turett [7] showed that forevery equivalent renorming ‖|·‖| of (c0, ‖·‖∞), there ex-ists a closed, bounded, convex set C and an asymptot-ically nonexpansive mapping T : C → C [i.e., there ex-ists a sequence (kn)n∈N in [1,∞) such that kn −→

n1, and

for all n ∈ N, for all x, y ∈ C, ‖|Tnx−Tny‖| ≤ kn‖|x−y‖|]such that T has no fixed point.

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In contrast to this, note that in 1972, Goebel andKirk [9] proved that for all uniformly convex spaces(X, ‖ · ‖), for every closed, bounded, convex set C ⊆ X,for all eventually asymptotically nonexpansive mapsT : C → C, T has a fixed point in C.In a recent paper of Lennard and Nezir [12] (Non-

linear Analysis 95 (2014), 414-420), using the above-described theorem of Domınguez Benavides and theStrong James’ Distortion Theorems, we proved thatif a Banach space is a Banach lattice, or has an un-conditional basis, or is a symmetrically normed idealof operators on an infinite-dimensional Hilbert space,then it is reflexive if and only if it has an equivalentnorm that has the fixed point property for cascadingnonexpansive mappings (see Definition 2.1).

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This new class of mappings strictly includes nonex-pansive mappings. Cascading nonexpansive mappings areanalogous to asymptotically nonexpansive mappings, butexamples show that neither of these two classes ofmappings contain the other. One of these examplesis due to Lukasz Piasecki. He invented an asymptoti-cally nonexpansive map that is not cascading nonex-pansive. This example will be described in the papercurrently being prepared by Lennard, Nezir and Pi-asecki [LNP].Note that via Lin’s example (described above) of

an equivalent renorming ‖| · ‖| of (`1, ‖ · ‖1) such that(`1, ‖|·‖|) has the fixed point property for nonexpansive

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mappings, in our theorem one cannot replace “cascad-ing nonexpansive mappings” by “nonexpansive map-pings”.

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2. Reflexivity iff the perturbed cascadingnonexpansive FPP in Banach lattices

Let C be a closed bounded convex subset of a Banachspace (X, ‖ · ‖). Let T : C −→ C be a mapping. LetC0 := C and

C1 := co(T (C)

)⊆ C .

Clearly C1 is a closed bounded convex set in C. Letx ∈ C1. Then

Tx ∈ T (C1) ⊆ T (C) ⊆ co (T (C)) = C1 .

So, T maps C1 into C1. Inductively, for all n ∈ N wedefine

Cn := co(T (Cn−1)

)⊆ Cn−1 .

Similarly to above, it follows that T maps Cn into Cn.

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Definition 2.1. Let (X, ‖ · ‖) be a Banach space and C be aclosed bounded convex subset of X . Let T : C −→ C be amapping and (Cn)n∈N be defined as above. We say that T iscascading nonexpansive if there exists a sequence (λn)n∈N in[1,∞) such that λn −→

n1, and for all n ∈ N, for all x, y ∈ Cn,

‖Tx− Ty‖ ≤ λn ‖x− y‖ .

Note that every cascading nonexpansive mapping isnorm-to-norm continuous; and every nonexpansive mapis cascading nonexpansive.

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Cascading nonexpansive mappings arise naturally inBanach spaces (X, ‖ · ‖) that contain an isomorphiccopy of `1 or c0. Examples of such spaces are Ba-nach spaces isomorphic to a nonreflexive Banach lat-tice, and nonreflexive Banach spaces with an uncon-ditional basis. (See, for example, Lindenstrauss andTzafriri [15] 1.c.5 and [14] 1.c.12.) Another class ofsuch spaces are Banach spaces isomorphic to a nonre-flexive symmetrically normed ideal of operators on aninfinite-dimensional Hilbert space. (See Peter Doddsand Lennard [4].)

Theorem 2.2. Let (X, ‖ · ‖) be a Banach space that containsan isomorphic copy of `1 or c0. Then there exists a closedbounded convex set C ⊆ X and an affine cascading nonexpan-sive mapping T : C −→ C such that T is fixed point free.

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The proof uses the Strong James’ Distortion Theo-rem for `1 and c0 ([6], [7]).

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Theorem 2.3. Let (X, ‖ · ‖) be a reflexive Banach space. Thenthere exists an equivalent norm ‖·‖∼ on X such that for everyclosed bounded convex subset C of X, for all ‖ · ‖∼-cascadingnonexpansive mappings T : C −→ C, T has a fixed point inC.

Proof. By Domınguez Benavides [5], there exists an equivalentnorm ‖ · ‖∼ on X such that for every closed bounded convex sub-set E of X , for all ‖ · ‖∼-nonexpansive mappings U : E −→ E,U has a fixed point in E. Fix an arbitrary closed bounded convexsubset C of X . Let T : C −→ C be a ‖ · ‖∼-cascading nonexpan-sive mapping. As above, let C0 := C and Cn := co

(T (Cn−1)

),

for all n ∈ N. By hypothesis there exists a sequence (λn)n∈N in[1,∞) such that λn −→

n1, and for all n ∈ N, for all x, y ∈ Cn,

‖Tx− Ty‖∼ ≤ λn ‖x− y‖∼.

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Now X is reflexive, and so C is weakly compact. By Zorn’sLemma there exists a closed bounded convex (non-empty) setD ⊆C such that D is a minimal invariant set for T . I.e., T (D) ⊆ D,and if F is a closed bounded convex (non-empty) subset of D withT (F ) ⊆ F , then F = D. It follows that co

(T (D)

)= D. Let

D0 := D and Dn := co(T (Dn−1)

), for all n ∈ N. Inductively,

we see that for all n ∈ N, D = Dn ⊆ Cn. Therefore, by ourhypotheses on T , for all x, y ∈ D, for all n ∈ N,

‖Tx− Ty‖∼ ≤ λn ‖x− y‖∼ .

But λn −→n

1. Consequently, for all x, y ∈ D,

‖Tx− Ty‖∼ ≤ ‖x− y‖∼ ;

i.e., T is ‖ · ‖∼-nonexpansive on D. By Domınguez Benavides [5],T has a fixed point in D ⊆ C. �

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Combining Theorem 2.2, the remarks preceding thattheorem, and Theorem 2.3, we get the following “fixedpoint property” characterization of reflexivity in Ba-nach lattices, Banach spaces with an unconditionalbasis, and symmetrically normed ideals of operatorson an infinite-dimensional Hilbert space.

Theorem 2.4. [Lennard and Nezir [12]] Let (X, ‖ · ‖) be aBanach lattice, or a Banach space with an unconditional basis,or a symmetrically normed ideal of operators on an infinite-dimensional Hilbert space. Then the following are equivalent.

(1) X is reflexive.

(2) There exists an equivalent norm ‖ · ‖∼ on X such thatfor all closed bounded convex sets C ⊆ X and for all ‖ · ‖∼-cascading nonexpansive mappings T : C −→ C, T has a fixedpoint in C.

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3. The FPP in general B-spaces and mappings ofcascading nonexpansive type

Definition 3.1. Let (X, ‖ · ‖) be a Banach space and C be aclosed bounded convex subset of X . Let T : C −→ C be amapping and (Cn)n∈N0

be defined as above. We say that T isof cascading nonexpansive type if T is norm-to-norm continuousand

limsupm−→∞

supx∈Cm

limsupµ−→∞

supz∈Cµ

[‖Tx− Tz‖ − ‖x− z‖

]≤ 0 .

Note that every nonexpansive map is cascading nonexpan-sive, every cascading nonexpansive mapping is of cascadingnonexpansive type, and every mapping of cascading nonex-pansive type is norm-to-norm continuous.

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Remark 3.2. There exists a mapping of cascading nonexpansivetype that is not cascading nonexpansive.

4. Reflexivity iff the perturbed FPP for mappingsof cascading nonexpansive type

Theorem 4.1 (Lennard, Nezir, Piasecki, in preparation). Let(X, ‖ · ‖) be a Banach space. Then the following are equiva-lent.

(1) X is reflexive.

(2) There exists an equivalent norm ‖ · ‖∼ on X such that forall closed bounded convex sets C ⊆ X and for all mappingsT : C −→ C of ‖ · ‖∼-cascading nonexpansive type, T has afixed point in C.

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5. Part (2) of my talk

In the recent paper of Burns, Lennard and Sivek(Studia Mathematica 223(3) (2014), 275-283), we provethe existence of a contractive and fixed point freemapping on a weakly compact convex subset of theBanach space L1[0, 1] (with its usual norm), which an-swers a long-standing open question. This work con-stitutes part of the doctoral dissertation of the thirdauthor [Siv].In 1965 Kirk [K] proved that every nonexpansive map-

ping U on a weakly compact convex subset C of a Ba-nach space X with normal structure has a fixed point,extending the analogous results of Browder [B1, B2]and Gohde [G] for uniformly convex spaces.

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For a long time it was unknown if every nonexpansivemapping U on a weakly compact convex subset C ofa Banach space X has a fixed point. In 1981 Alspach[A] settled this open question by inventing the firstexample of a nonexpansive mapping T on a weaklycompact convex C in a Banach space X for which Tis fixed point free. Alspach’s mapping is an isometry,and X = L1[0, 1], with its usual norm. Soon after, Sine[Si] and Schechtman [Sc] invented more of these inter-esting fixed point free isometries T (again on a weaklycompact convex C ⊆ X = an L1-space, with its usualnorm).It is easy to check that for Alspach’s mapping T ,S := (I + T )/2 is another nonexpansive fixed point free

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map on C. Moreover, S contracts the distance be-tween some pairs of unequal points and preserves thedistance between other such pairs. Further, this factis true for S when T is Sine’s map. We thank BraileySims for pointing out to us that this is also true for Swhen T is any one of Schechtman’s mappings.The question as to whether there exists a contractive

mapping U (i.e., U contracts the distances betweenall pairs of unequal points) that is fixed point free ona weakly compact convex subset of a Banach spacewas still open. This question remained open until theauthors recently resolved it (see Theorem 5.1 below).

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We now describe this solution. First, we define theset

C1/2 =

{f : [0, 1]→ [0, 1] : f ∈ L1[0, 1] and

∫ 1

0f =

1

2

}.

This set is a weakly compact convex subset of theLebesgue function space L1[0, 1], with its usual norm‖ · ‖1. For the rest of Part (2) of this talk, T will standfor Alspach’s map as defined in [A].Alspach’s mapping T is given by: for all integrable

functions f : [0, 1] −→ [0, 1],

(Tf )(x) =

{2f (2x) ∧ 1 , 0 ≤ x < 1

2(2f (2x− 1) ∨ 1)− 1 , 1

2 ≤ x < 1.

Here, for all α, β ∈ R, α ∧ β := min{α, β} and α ∨ β :=max{α, β}.

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Alspach’s map preserves areas in the sense that ‖Tf−Tg‖1 = ‖f − g‖1 for all integrable functions f, g : [0, 1]→[0, 1]. In particular T : C1/2 −→ C1/2. This and otherfacts about Alspach’s mapping were discussed in Alspach[A]; and also in, for example, Day and Lennard [DL](where the minimal invariant sets of T are character-ized). In the above-mentioned paper of Burns, Lennardand Sivek, we proved the following theorem.

Theorem 5.1. The mapping

R : C1/2→ C1/2 : f 7→∞∑n=0

Tnf

2n+1=

(I

2+T

4+T 2

8+ · · ·

)(f )

is contractive [i.e., ‖Rf − Rg‖1 < ‖f − g‖1, for all f 6= g inC1/2] and fixed point free on C1/2.

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6. Part(3) of my talk: Introduction.

Recently, T. Gallagher, C. Lennard and R. Popescu(J. Mathematical Analysis and Applications 431 (2015),471-481) showed that there exists a non-weakly com-pact, closed, bounded, convex (c.b.c.) subset W of theBanach space of convergent scalar sequences (c, ‖·‖∞),such that every nonexpansive mapping T : W −→ Whas a fixed point. This answers a question left openin the 2003 and 2004 papers of Dowling, Lennard andTurett [DLT1], [DLT2].Consider the related Banach space of all scalar se-

quences that converge to zero, (c0, ‖ · ‖∞). In [DLT2]Theorem 1, the authors show that for a c.b.c. subsetK of (c0, ‖ · ‖∞) to be weakly compact, it is sufficientfor every nonexpansive mapping U : K −→ K to have

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a fixed point. Earlier, Maurey [Maur] had shown thatthis condition is also necessary; and consequently itcharacterizes weak compactness among the c.b.c. sub-sets of (c0, ‖ · ‖∞).In [DLT1] Theorem 4, the authors prove the following

preliminary theorem: [Every non-weakly compact, closed,bounded, convex subset K of (c0, ‖ · ‖∞) contains a furthersubset C of the same type for which there exists a nonexpan-sive mapping V : C −→ C that is fixed point free]. In [DLT1]Corollary 7, they proved the analogous result in thespace (c, ‖·‖∞): [Every non-weakly compact, closed, bounded,convex subset K of (c, ‖ · ‖∞) contains a further subset C ofthe same type for which there exists a nonexpansive mappingV : C −→ C that is fixed point free]. It was left open as to

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whether this result could be extended to: (‡) [On ev-ery non-weakly compact, closed, bounded, convex subset K of(c, ‖ · ‖∞) there exists a nonexpansive mapping U : K −→ Kthat is fixed point free]. That the analogous strengthen-ing is true in (c0, ‖ · ‖∞) was established in [DLT2], asmentioned above.The main purpose of the above-mentioned paper is

to show via appropriate examples that statement (‡) isfalse. (See Theorem 8.1, and many other examples inour paper...) These are also the first examples of non-weakly compact, closed, bounded, convex subsets Dof a Banach space X isomorphic to c0, such that D hasthe fixed point property for nonexpansive mappings.It is an interesting related fact, proven by Maurey

[Maur] (also see Borwein and Sims [Borw-Sims]), that the

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analogue of Maurey’s theorem in c0 is true in (c, ‖·‖∞):[For all weakly compact, convex subsets E of (c, ‖ · ‖∞), everynonexpansive mapping T : E −→ E has a fixed point].In Section 9 we use Theorem 8.1 and the fact that

(c, ‖·‖∞) is isomorphic to (c0, ‖·‖∞), to define an equiv-alent norm ‖·‖∼ on c0 for which there exist non-weaklycompact c.b.c. subsets that have the fixed point prop-erty (FPP) for ‖ · ‖∼-nonexpansive mappings. (SeeTheorem 9.1.)Finally, as we remarked above, Lin [Lin] proved that

for a certain equivalent norm ‖| · ‖| on `1, every c.b.c.subset C of `1 has the FPP. Whether an analogous suchequivalent norm exists on c (or equivalently, c0) re-mains an open question.

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7. Part (3): Preliminaries.

The Banach space of all bounded sequences (`∞, ‖·‖∞)is given by

`∞ :=

{x = (xn)n∈N : xn ∈ R , and ‖x‖∞ := sup

n∈N|xn| <∞

}.

The Banach space of all convergent sequences (c, ‖·‖∞)is defined by

c :={x = (xn)n∈N : xn ∈ R , and λ(x) := lim

n−→∞xn exists in R

}.

Moreover, the Banach space of all convergent to zerosequences(c0, ‖ · ‖∞) is defined by

c0 :={x = (xn)n∈N : each xn ∈ R , and lim

n−→∞xn = 0

}.

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Clearly, each of the above Banach spaces contains thenext as a proper closed subspace. We also define c+

0 :={x ∈ c0 : each xj ≥ 0}.Following Aronszajn and Panitchpakdi [AP] Defini-

tion 1, p. 410 (also see, for example, Goebel andKirk [Goeb-Kirk] Definition 4.4, pp. 46-47), we say thata metric space (M,ρ) is hyperconvex if for all families(B(xa; ra)

)a∈A of closed balls in M for which

ρ(xa, xb) ≤ ra + rb , for all a, b ∈ A ,

it follows that ⋂a∈A

B(xa; ra) 6= ∅ .

Examples of hyperconvex metric spaces include thereal line with its usual metric, (R, d|·|); (Rn, d‖·‖∞), for

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all n ∈ N; and (`∞, d‖·‖∞). Also, all closed balls and order

intervals in these metric spaces are hyperconvex (see,for example, [Goeb-Kirk], p. 47 and p. 49).

We will use the following interesting theorem of Soardi[Soardi]. (Also see, for example, [Goeb-Kirk] p. 48.)

Theorem 7.1 (Soardi, 1979). Let (M,ρ) be a bounded, hy-perconvex metric space. Then every nonexpansive mappingT : M −→M has a fixed point.

We will also use the interesting characterization the-orem below that is a corollary of Aronszajn and Pan-itchpakdi [AP] Theorem 9, p. 423. (Also see, for ex-ample, Espınola and Khamsi [Kirk-Sims] Corollary 4.8,p. 401.) A mapping R from a metric space (M,ρ) into

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a metric subspace S is called a retraction of M onto S,if R is continuous on M and R(s) = s, for all s ∈ S.

Theorem 7.2 (Aronszajn and Panitchpakdi, 1956). Let (H, d)be a metric space. Then the following are equivalent.(1) H is hyperconvex.(2) There exists a hyperconvex metric space (M,ρ) such thatH ⊆ M and ρ|H×H = d, and a nonexpansive retraction R ofM onto H.

8. A non-weakly compact c.b.c. subset of c withthe fixed point property.

Theorem 8.1. There exists a non-weakly compact, closed,bounded, convex subset W of the Banach space (c, ‖ ·‖∞) suchthat every nonexpansive mapping T : W −→ W has a fixedpoint.

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Proof. Let us define

W := {y = (yn)n∈N ∈ `∞ : 1 ≥ y1 ≥ y2 ≥ y3 ≥ · · · ≥ 0} .Clearly, by the monotone convergence theorem, W is a subset ofc. It is easy to check that W is a closed, bounded, convex subsetof (c, ‖ · ‖∞). That W is not weakly compact may be seen byusing the criterion for weak convergence of sequences in c given inBanach [Ban] Chapter IX, p. 83.We will show that the metric space (W,d‖·‖∞) is hyperconvex.

The theorem will then follow from Soardi’s Theorem [Soardi] (seeTheorem 7.1 above).Let 0 := (0, 0, 0, . . . , 0, . . . ) and 1 := (1, 1, 1, . . . , 1, . . . ). Con-

sider the order interval L := [[0,1]] := {u = (un)n∈N : 0 ≤un ≤ 1 , for alln ∈ N}. Then (L, d‖·‖∞) is a hyperconvex metric

space. (See, for example, [Goeb-Kirk] Remark 4.1, p. 49.) Alsonote that W is a subset of L.

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We define the mapping S : L −→ W by

S(y) := (y1, y1∧y2, y1∧y2∧y3, . . . ) , for all y = (yj)j∈N ∈ L .

The map S is clearly a retraction of L onto W . Moreover, it iseasy to check that

‖S(x)− S(y)‖∞ ≤ ‖x− y‖∞ , for all x, y ∈ L .

Thus, S is a nonexpansive retraction of the hyperconvex metricspace L onto W . Hence (W,d‖·‖∞) is hyperconvex, and so hasthe the fixed point property for nonexpansive mappings. �

9. An equivalent norm on c0 s.t. some non-weaklycompact, c.b.c. sets have the fixed point

property.35

We define the equivalent norm ‖ · ‖∼ on c0 by theformula

‖β‖∼ := maxk∈N|βk+1 + β1| , for all β = (βj)j∈N ∈ c0 .

We further define the non-weakly compact, closed,bounded, convex subset K of c0 by

K :={α = (α1, α2, . . . , αj, . . . ) ∈ c+

0 :

α1 + α2 ≤ 1 and α2 ≥ α3 ≥ α4 ≥ . . .}.

Theorem 9.1. The non-weakly compact, closed, bounded, con-vex subset K of c0 has the fixed point property for ‖ · ‖∼-nonexpansive mappings.

Thank you !

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Proof. Consider the one-to-one continuous linear mapping U :c −→ c0 given by

U : u = (u1, u2, u3, . . . ) 7→ (λ(u), u1−λ(u), u2−λ(u), u3−λ(u), . . . ) .

The map U is clearly onto and the inverse mapping U−1 : c0 −→c is given by

U−1 : β = (β1, β2, β3, . . . ) 7→ (β2 + β1, β3 + β1, β4 + β1, . . . ) .

Clearly, U is a linear isomorphism of (c, ‖ · ‖∞) onto (c0, ‖ · ‖∞).Moreover, by its construction, U is a linear isometric isomorphismof(c, ‖ · ‖∞) onto (c0, ‖ · ‖∼). Recall that the following non-weaklycompact, closed, bounded, convex subset of (c, ‖ · ‖∞) has thefixed point property for nonexpansive mappings:

W := {y = (yn)n∈N ∈ `∞ : 1 ≥ y1 ≥ y2 ≥ y3 ≥ · · · ≥ 0} .It is easy to check that U(W ) = K. �

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Thank you again !

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